Characterizations of semiperfect and perfect rings

Publicacions Matem`atiques, Vol 40 (1996), 115–125. CHARACTERIZATIONS OF SEMIPERFECT AND PERFECT RINGS(∗) Weimin Xue Abstract We characterize semiperfect modules, semiperfect rings, and perfect rings using locally projective covers and generalized locally projective covers, where locally projective modules were introduced by Zimmermann-Huisgen and generalized locally projective coves are adapted from Azumaya’s generalized projective covers. Introduction Azumaya [A2] introduced the notion of generalized projective covers to characterize semiperfect modules and rings. Adapting his concept, we call a module epimorphism f:P−→ Ma(generalized)cover in case (Ker(f)⊆Rad(P)) Ker(f)P. A (generalized) cover f:P−→ Mis called a (generalized)projective cover in case Pis a projective module, and it is called a (generalized)locally projective cover in case Pis a locally projective module. This paper consists of three sections. We obtain some basic properties of (generalized) covers in Section 1. In Section 2, we characterize (generalized) semiperfect modules via (generalized) projective covers of the (generalized) complements. In Section 3, we characterize semiperfect rings and modules, perfect rings, and quasi-perfect rings [CX], using (generalized) locally projective covers. The terminologies and notations of Anderson and Fuller [AF] will be freely used. We refer the reader to [AF, Section 27, Section 28] for a presentation of semiperfect and perfect rings. Throughout Ris an associative ring with identity whose Jacobson radical is denoted by J. Unless otherwise stated, modules are unitary left R-modules, and homomorphisms are left R-module homomorphisms. If Mis a module, we recall from [AF] that Rad(M) denotes the radical of M, and (U M)U≤Mmeans that Uis a (superfluous) submodule of M. (∗)This research is supported by the National Natural Science Foundation of China. 116 W. Xue 1. Basic properties of (generalized) covers If Pand Mare modules, we call an epimorphism f:P−→ Ma (generalized)cover in case (Ker(f)⊆Rad(P)) Ker(f)P. Since Rad(P) is the sum of all superfluous submodules of P, every cover is a generalized cover. We have the following basic properties of (generalized) covers. Lemma 1.1. If both f:P−→ Mand g:M−→ Nare (generalized) covers, then gf :P−→ Nis a (generalized) cover. Proof: If both fand gare covers, then gf isacoverby[AF, Proposition 5.17(1)]. Now let both fand gbe generalized covers. To show Ker(gf)⊆ Rad(P), we let p∈Ker(gf). Then gf(p)=0andf(p)⊆Ker(g)⊆ Rad(M). Since Ker(f)⊆Rad(P), it follows from [AF, Proposition 9.15] that f(Rad(P)) = Rad(M). Hence f(p)=f(p) for some p∈Rad(P), so p−p∈Ker(f)⊆Rad(P). We obtain p∈Rad(P). Lemma 1.2. (1) If each fi:Pi−→ Mi(i=1,2,... ,n)is a cover then ⊕n i=1fi:⊕n i=1Pi−→ ⊕ n i=1Miis a cover. (2) If each fi:Pi−→ Mi(i∈I)is a generalized cover then ⊕i∈Ifi: ⊕i∈IPi−→ ⊕ i∈IMiis a generalized cover. Proof: (1) Since each Ker(fi)Piwe have Ker(⊕n i=1fi)= ⊕n i=1 Ker(fi)⊕ n i=1Pi.So⊕n i=1fiis a cover. (2) Since each Ker(fi)⊆Rad(Pi) we have Ker(⊕i∈Ifi)=⊕i∈IKer(fi)⊆ ⊕i∈IRad(Pi) = Rad(⊕i∈IPi). So ⊕i∈Ifiis a generalized cover. Lemma 1.3. Let f:P−→ Mbe a cover. If Mis finitely generated then Pis also finitely generated. Proof: Since P/Ker(f)∼ =Mis finitely generated, there is a finitely generated submodule Pof Psuch that P+ Ker(f)=P. Since Ker(f)Pwe have P=P. Semiperfect and perfect rings 117 2. (Generalized) projective covers and M-projective covers Let Mbe a module. If U,U≤Mand M=U+Uthen Uis called a (generalized)complement of Uin case (U∩U⊆Rad(U)) U∩UU. Clearly, each complement is a generalized complement. A (generalized) cover f:P−→ Mis called a (generalized)projective cover in case Pis a projective module. Since every cover is a generalized cover, a projective cover is a generalized projective cover as observed in [A2]. A connection between (generalized) projective covers and (generalized) complements is given as follows. Proposition 2.1. If Mis a module and U≤M, then the following three statements are equivalent. (1) M/U has a (generalized) projective cover. (2) If V≤Mand M=U+Vthen Uhas a (generalized) complement U⊆Vsuch that Uhas a (generalized) projective cover. (3) Uhas a (generalized) complement Uwhich has a (generalized) projective cover. Proof: (1) ⇒(2). Let f:P−→ M/U be a (generalized) projective cover. Since M=U+V, g:V−→ M/V via v−→ v+U is an epimorphism. Since Pis projective, there is a homomorphism h:P−→ Vsuch that f=gh. It is easy to see that M=U+h(P) where h(P)⊆V. Now (Ker(f)⊆Rad(P)) Ker(f)P,sowehave U∩h(P)=h(Ker(f)) (⊆h(Rad(P)) ⊆Rad(h(P))) h(P) and h(P) is a (generalized) complement of U⊆V. Since Ker(h)⊆ Ker(f)(⊆Rad(P)) P, h:P−→ h(P) is a (generalized) projective cover. (2) ⇒(3). This is obvious. (3) ⇒(1). Let f:P−→ Ube a (generalized) projective cover. Since Uis a (generalized) complement of U, the natural epimorphism g:U−→ U/(U∩U) h ∼ =(U+U)/U =M/U 118 W. Xue is a (generalized) cover. Hence hgf :P−→ M/U is a (generalized) projective cover by Lemma 1.1. Let Mbe a module. Mis called (generalized)complemented in case each submodule Uhas a (generalized) complement U, and it is called (generalized)amply complemented in case M=U+Vimplies that U has a (generalized) complement U⊆V. According to [A2], Mis called (generalized)semiperfect in case each factor module of Mhas a (generalized) projective cover. Azumaya [A2, Theorem 4] proved that M is generalized semiperfect if and only if each proper submodule of Mis contained in a maximal submodule of Mand each simple factor module of Mhas a generalized projective cover. The next different characterization of (generalized) semiperfect modules follows immediately from Proposition 2.1, where the non-parenthetical version is [F, Theorem 1]. A characterization of semiperfect modules using locally projective covers will be given in the next section. Theorem 2.2. The following three statements are equivalent for a module M. (1) Mis (generalized) semiperfect. (2) Mis (generalized) amply complemented by complements which have (generalized) projective covers. (3) Mis (generalized) complemented by complements which have (generalized) projective covers. A (generalized) cover f:P−→ Mis called a (generalized)Mprojective cover in case Pis a M-projective module. Modifying the proof of Proposition 2.1, we have an analogous result. Proposition 2.3. If Mis a module and U≤M, then the following three statements are equivalent. (1) M/U has a (generalized) M-projective cover. (2) If V≤Mand M=U+Vthen Uhas a (generalized) complement U⊆Vsuch that Uhas a (generalized) M-projective cover. (3) Uhas a (generalized) complement Uwhich has a (generalized) M-projective cover. We call a module M(generalized)quasi-semiperfect in case each factor module of Mhas a (generalized) M-projective cover. Now we have the following result by Proposition 2.3. Theorem 2.4. The following three statements are equivalent for a module M. (1) Mis (generalized) quasi-semiperfect. Semiperfect and perfect rings 119 (2) Mis (generalized) amply complemented by complements which have (generalized) M-projective covers. (3) Mis (generalized) complemented by complements which have (generalized) M-projective covers. Using an idea of Azumaya’s proof given in [A2, Theorem 4] we obtain Theorem 2.5. Let Rbe a semilocal ring and Ma finitely generated left R-module. Then Mis (generalized) quasi-semiperfect if and only if each simple factor module of Mhas a (generalized) M-projective cover. Proof: (⇒). This is obvious. (⇐). Let U≤Mand M=M/U. Since Ris semilocal and M is finitely generated, JM= Rad(M)Mand M/JMis semisimple. Let M/JM=⊕n i=1Sibe a direct sum of simple submodules Si (i=1,2,... ,n). Since each Siis isomorphic to a simple factor module of M, it has a (generalized) M-projective cover fi:Pi−→ Siwhere (Ker(fi)⊆Rad(Pi)=JPi) Ker(fi)Pi. Since Pi/Ker(fi)∼ =Siis simple, Ker(fi) is a maximal submodule of Pand so (Ker(fi)=JPi) Ker(fi)=JPiPi. By Lemma 1.2, f=⊕n i=1fi:P=⊕n i=1Pi−→ ⊕ n i=1Si=M/JM is a (generalized) M-projective cover, where (Ker(f)=JP) Ker(f)= JP P. Let g:M−→ M/JMbe the natural epimorphism. Since Pis M-projective, there is a homomorphism h:P−→ Msuch that f=gh. Now fis an epimorphism and Ker(g)=JMM, it follows from [AF, Corollary 5.15] that his an epimorphism. Since (Ker(h)⊆Ker(f)=JP) Ker(h)⊆Ker(f)=JP P, we see that h:P−→ M is a (generalized) M-projective cover. Hence Mis a (generalized) quasisemiperfect module. 3. (Generalized) locally projective covers A module Pis called locally projective [Z1] in case it satisfies any of the following equivalent conditions: (a) if Aand Bare modules, g:A−→ B is an epimorphism and f:P−→ Bis a homomorphism then for every finitely generated (cyclic) submodule P0of Pthere is a homomorphism h:P−→ Asuch that f|P0=gh|P0; (b) if Mis a module and f:M−→ 120 W. Xue Pis an epimorphism then for every finitely generated (cyclic) submodule P0of Pthere is a homomorphism g:P−→ Msuch that fg|P0=1 P0. Clearly, every finitely generated (even countably generated [A1]) locally projective module is projective. The following facts are also known and we shall freely use them without reference (for the proofs, see [Z1] and [A1]): (1) a direct sum of modules is locally projective if and only if each summand is locally projective; (2) a pure submodule of a locally projective module is locally projective; and (3) if Pis a locally projective module, then (i) Pis flat, (ii) Rad(P)=JP, and (iii) if Rad(P)=P then P=0. A (generalized) cover f:P−→ Mis called a (generalized)locally projective cover in case Pis a locally projective module. Since any cover is a generalized cover, a locally projective cover is a generalized locally projective cover. According to the facts of locally projective modules and Lemmas 1.1, 1.2 and 1.3, we have the following three lemmas. Lemma 3.1. If f:P−→ Mis a (generalized) locally projective cover and g:M−→ Nis a (generalized) cover then gf :P−→ Nis a (generalized) locally projective cover. Lemma 3.2. (1) If each fi:Pi−→ Mi(i=1,2,... ,n)is a locally projective cover then ⊕n i=1fi:⊕n i=1Pi−→ ⊕ n i=1Miis a locally projective cover. (2) If each fi:Pi−→ Mi(i∈I)is a generalized locally projective cover then ⊕i∈Ifi:⊕i∈IPi−→ ⊕ i∈IMiis a generalized locally projective cover. Lemma 3.3. Let f:P−→ Mbe a locally projective cover. If Mis a finitely generated module then Pis a finitely generated projective module. The following proposition is an analogous result of [A2, Proposition 1]. Proposition 3.4. Let f:P−→ Mbe a generalized locally projective cover. If g:Q−→ Mis a projective cover where Qis finitely generated then there is an isomorphism h:P∼ =Qsuch that f=gh. Proof: Let Q=n i=1 Rqi. Then f(pi)=g(qi) for some pi∈P (i=1,2,... ,n). Since P0=n i=1 Rpiis a finitely generated submodule of Pthere is a homomorphism h:P−→ Qsuch that f|P0=gh|P0. Now g(qi)=f(pi)=gh(pi), so qi−h(pi)∈Ker(g) and we have Q= h(P0)+Ker(g). But Ker(g)Qwe obtain h(P0)=Q,soh(P)=Qand his an epimorphism. Since Qis projective, hsplits. Let K= Ker(h)⊆ Ker(f) and P=K⊕Kfor some K≤P. Since f:P−→ Mis Semiperfect and perfect rings 121 a generalized locally projective cover, we have Ker(f)⊆JP and then K⊆JP =JK ⊕JK. It follows that K=JK. Since Kis locally projective, we get K= 0, i.e., his a monomorphism. Thus his an isomorphism. Recall that Ris semiperfect if R/J is semisimple and idempotents lift modulo J. It is known that Ris semiperfect if and only if every simple (finitely generated, cyclic) left R-module has a projective cover (see, e.g. [AF, Theorem 27.6]). Azumaya [A2, Theorem 3] generalized this and proved that if every simple left R-module has a generalized projective cover then Ris semiperfect. Modifying his proof we generalize his theorem as follows. Theorem 3.5. The following three statements are equivalent for a ring R. (1) Ris semiperfect. (2) Every simple left R-module has a locally projective cover. (3) Every simple left R-module has a generalized locally projective cover. Proof: (1) ⇒(2) ⇒(3). These are clear. (3) ⇒(1). To show R=R/J is semisimple, we only need to prove each simple left R-module Sis locally projective (since a simple locally projective module is projective). We regard Sas a simple left R-module, so there is a generalized locally projective cover f:P−→ S, where Ker(f)⊆Rad(P)=JP. Since Ker(f) is a maximal submodule of P we must have Ker(f)=JP and so P/JP ∼ =S. Since Pis a locally projective R-module, P/JP is a locally projective R-module, so Sis a locally projective R-module. Therefore Ris semisimple. Let εbe an idempotent of R. We want to show that εcan be lifted to an idempotent of R, so we may assume that ε=0 and ε= 1. Then both Rε and R(1 −ε) are non-zero left ideals of the semisimple ring R. Let Rε =S1⊕···⊕Skand R(1 −ε)=Sk+1 ⊕···⊕Snbe direct sums of simple left ideals Si’s. We view Sias a simple left R-module and let fi:Pi−→ Sibe a generalized locally projective cover (i=1,2,... ,n). Then f=⊕n i=1fi:P=⊕n i=1Pi−→ ⊕ n i=1Si is a generalized locally projective cover by Lemma 3.2. Since the natural epimorphism g:R−→ Ris a projective cover, by Proposition 3.4 there is an isomorphism h:P−→ Rsuch that f=gh. Let L=h(P1⊕···⊕Pk) and L=h(Pk+1 ⊕···⊕Pn). Then Land Lare left ideals of Rand 122 W. Xue R=L⊕L. Let L=Re and L=Refor some idempotents eand eof Rwith e+e= 1. Let e=g(e)∈R. Then Re =g(Re)=g(L)=gh(P1⊕···⊕Pk)=f(P1⊕···⊕Pk) =f1(P1)⊕···⊕fk(Pk)=S1⊕···⊕Sk=Rε. Similarly, if we let e=g(e)∈Rthen Re=R(1 −ε). Now 1=g(1) = g(e+e)=e+e.By[AF, Proposition 7.2] we must have ε=e, i.e., ε can be lifted to the idempotent e∈R. Corollary 3.6. The following three statements are equivalent for a ring R. (1) Ris semiperfect. (2) Every finitely generated (cyclic) left R-module has a locally projective cover. (3) Every finitely generated (cyclic) left R-module has a generalized locally projective cover. Next we characterize semiperfect modules using locally projective covers, but we need a lemma first. Lemma 3.7. If a module Mhas a generalized locally projective cover then (1) Rad(M)=JM; and (2) Mhas a maximal submodule if M=0. Proof: Let f:P−→ Mbe a generalized locally projective cover. Then Ker(f)⊆Rad(P)=JP.By[AF, Proposition 9.15], we have Rad(M)=f(Rad(P)) = f(JP)=J(f(P)) = JM.IfM= 0, then P= 0 and Rad(P)=P.SoPhas a maximal submodule U. Since Ker(f)⊆Rad(P)⊆U,f(U) must be a maximal submodule of f(P)= M. Proposition 3.8. A module Mis semiperfect if and only if Mhas a projective cover and every factor module of Mhas a locally projective cover. Proof: (⇒). This is obvious. (⇐). Let Ube a proper submodule of M. Then M/U is a non-zero factor module of M. By Lemma 3.7, M/U has a maximal submodule. This means that Uis contained in a maximal submodule of M.By assumption, every simple factor module of Mhas a locally projective cover which is a projective cover by Lemma 3.3. Hence Mis semiperfect by [A2, Theorems 4 and 6]. Semiperfect and perfect rings 123 Corollary 3.9. A projective module Mis semiperfect if and only if every factor module of Mhas a locally projective cover. Recall that Ris left perfect if every left R-module has a projective cover. An interesting characterization of left perfect rings was presented in [AF, Theorem 28.4] which was due to Bass [B]. Now we characterize left perfect rings using (generalized) locally projective covers. Theorem 3.10. The following three statements are equivalent for a ring R. (1) Ris left perfect. (2) Every left R-module has a locally projective cover. (3) Every left R-module has a generalized locally projective cover. Proof: (1) ⇒(2) ⇒(3). These are clear. (3) ⇒(1). By Theorem 3.5 or Corollary 3.6, Ris semiperfect. By Lemma 3.7, every non-zero left R-module has a maximal submodule. Hence Ris left perfect by [AF, Theorem 28.4]. It is known that if every semisimple left R-module has a projective cover then Ris left perfect. We do not know whether the condition “projective cover” can be weakened to “locally projective cover”. Azumaya [A2, Theorem 1] showed that a flat module having a generalized projective cover is projective. An analogous result for locally projective modules is the following Proposition 3.11. If Mis a flat module having a generalized locally projective cover then Mis locally projective. Proof: Let f:P−→ Mbe a generalized locally projective cover and K= Ker(f). Then K⊆Rad(P)=JP.By[AF, Lemma 19.18] Kis a pure submodule of P,soKis also locally projective and JK =K∩JP ⊇ K. We get K=JK, and so K= 0. Hence P∼ =Mand Mis locally projective. As pointed out by Zimmermann-Huisgen in [Z2, p. 60], Ris left perfect if and only if every flat left R-module is locally projective. Hence by Proposition 3.11 we obtain Corollary 3.12. A ring Ris left perfect if and only if every flat left R-module has a (generalized) locally projective cover. Camillo and Xue [CX] called a ring Rleft quasi-perfect in case every artinian left R-module has a projective cover, and showed that the class